MAT 1033 Name College Algebra with Calculus for Business COMMUNICATION OBJECTIVE

Question

MAT 1033 Name College Algebra with Calculus for Business COMMUNICATION OBJECTIVE QUIZ Quadratic Equations & Functions Using pencil or pen, hand-PRINT all the requested answers as specified by the questions. 1.) Find the quadratic function f in standard form whose graph has its vertex at (3, 2) and passes through the point (2, 1). On the left column, write every Algebra step required to find t equation. On the right column, hand print in words a detailed and well-written paragraph explaining what and why you are doing each Algebra step. Part 1.b) Part 1.a) Written Communication Print a paragraph that Visual Communication explains each step using complete sentences and Detailed Algebra Steps correct grammar Given any quadratic functionf If we let fx 0 and the solutions to the resulting qua uation are not real numbers, write a complete sentence explaining what that result tel out the x-intercepts of the graph of the function f?

Explanation / Answer

Standard form of quadratic function is y = ax2 + bx + c, and this function results in a parabola.
Vertex of the parabola (h,k) is such that h = (-b/2a)
So -b / 2a = -3
So b = 6a

Also the k = (4ac - b2 ) / 4a
So 2 = (4ac - 36a2) / 4a ----------------------------------------------[ because b = 6a ]
So 2 = 4c - 36a
So c = (2 + 36a) / 4

Putting the values of b and c in the original function.
y = ax2 + 6ax + (2 + 36a) / 4
So 4y = 4ax2 + 24ax + 36a + 2

Now since the parabola passes through (2,1), so our function will satisfy for the values x = 2 and y = 1

So 4(1) = 4a(2)2 + 24a(2) + 36a + 2
So 4 = 16a + 48a + 36a + 2
So 100a = 2
so a = 1 / 50
Therefore b = 6 / 50 = 3 / 25
And c = (2 + 36/50) / 4
c = 17 / 25

So y = 1/50 x2 + 3/25 x + 17/25
OR
y = ( x2 + 6x + 34 ) / 50



2)
Roots of a quadratic equation have a graphical interpretation: they're the places where the function crosses the x-axis. But this happens only when the roots are real.
When there are no real roots, the parabola (i.e. the graph of quadratic equation) lies entirely above x axis, thereby not making any intercept with the X-axis.

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